Fermi level energy

The Fermi level energy Ef appearing in eq. (2.16) [oreq. (2.17)] through the argument b... 

In view of the result just found, it is interesting to contrast exact and approximate behaviour of the density n(x F) [eqs. p.5) and (2.17), respectively]. Some insight into the nature of the approximations contained in our treatment is gained through the inspection of Table 1, which collects Fermi-level energy values calculated for several electron occupation numbers and two different electric field amplitudes. The entries have... 

Table 1. Fermi-level energy Ef predicted for the harmonic-well model = 1. a.M.) for different electron numbers (2iV ) and different electric field amplitudes E (a.u.).

From equation (3.4.11) we see that the energy gap between the conduction band edge and the Fermi energy level is a logarithmic function of donor concentration. As the donor concentration increases so does the electron concentration in the conduction band, with the Fermi level energy moving closer to the conduction band edge. 

Fig. 1.8 Plot of energy E in metallic silver against atomic radius r. (I) energy E0 of lowest state (II) mean Fermi energy f F of electron gas (III) mean energy E0 + EF of electron at the Fermi level Energy in rydbergs (13.52 eV). From Mott and Jones (1936).

All this material about the Fermi-Dirac equation for the probability of filling of the electron states comes down in practice to one approximation Electrons taking part in electrode processes are from the Fermi level and hence have the Fermi level energy. 

Figure 3. DOS for a) semiconductor BCN-NT (3,3) type-6 and b) metal BCN-NT (6,0) type-3 . The Fermi level energy is taken as zero.

The difference between the equihbrium Fermi level energy and the energy of the bottom of the conduction band at the semiconductor/metal interface is also an important quantity often mentioned in literature. This difference is called... 

Figure 2.3. Band diagram for titanium metal, illustrating the continuum between valence and conduction bands (i. e., no bandgap). The Fermi level corresponds to the highest occupied energy state at absolute zero (think of pouring coffee (electrons) into a mug (valence band) - the top of the fluid level represents the Fermi level/energy).

As referred to earlier, the usual representation of effects of electrode potential on electrochemical reaction rates is through a modulating term pVF operating on AG° [Eq. (11)]. [Taking account of doublelayer structure, this term is written as )8( V - il/i)F where il/i is the diffuse-layer potential, but this is a trivial difference in the present context.] Change of potential, V, modifies the Fermi level energy by eV as in Eq. (10). In the usual transition-state treatment, this quantity, modified by the factor p, appears in the Arrhenius-Boltzmann exponent [-(AG - iSVF)/RT]. It is from this exponent that the conventional Tafel-slope quantity b arises, linear in T. 

In the original treatment of Gurney/ the current was expressed as the integral of the product of electrolyte and electron energy distribution functions but with the electronic one written as a Boltzmann factor, exp( A /fcT). The symmetry factor was introduced intuitively in terms of the shift of intersection point of energy profiles in relation to change of electrode potential, i.e., of the Fermi-level energy (cf. Butler ). 

The question of how the p factor enters in relation to the applicability of the Fermi-Dirac distribution in electron transfer processes appears to require further exploration. The problem arises from the way change of Fermi-level energy is assumed just to shift the potential energy surface up or down by VF and hence cause a change of (1 - p)VF in the transition-state energy according to the Br0nsted principle. 

The potential-dependent exponential term is also independent of v. This may be explained if we recall that the effect of an applied overpotential in electrochemical kinetics is to modify the relative energies of reactant and product curves through a change in the Fermi level energy and, by extension, that of the transition state for the elanentary (recall footnote h) reaction step. The potential dependence of the energy of the transition state is sensitive to the number of electrons transferred in it, but not the v times that it must occur per turnover of the overall reaction thus the exponential in Eq. (52) is independent of v. 

Moreover, let us consider a large metal particle that has electronic levels so dose that they actually form bands. The spacing between adjacent levels is approximately expressed as 5=Ef/N where Cp is the Fermi level energy and N is the number of atoms in the particle [102]. As the spacing between the levels becomes larger than the thermal energy kT, (k Boltzmann constant) the levels begin to behave individually and the particle may lose its metallic properties. At room temperature kT= 2.5 x 10 eV, and with Ep of the order of 10 eV, N is calculated to be approximately 400, which corresponds to a diameter of about 2nm. This... 

In Eq. (10.1), Pjjj( ) is the surface electronic density of states. the interaction Hamiltonian, cp. the adsorbate atomic orbital, xp-g a metal surface orbital, and a the energy of an adsorbate orbital. The tight-binding overlap energy between adsorbate and surface atomic orbital is described as f and that between two metal atomic orbitals as f. As the adsorbate atomic orbital energy equals the Fermi level energy and metal orbitals are half-filled, a simple expression results for the interaction energy ... 

Hence, at equilibrium, —Vredox(vacuum scale) (which is known as Ep.redox) cannot be equal to Fermi level energy in metal or semiconductor, since -eoVredox(vacuum scale) is the chemical (and not the electrochemical) potential of electrons in the solution. 

---